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Theorem u5lemnana 649
Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemnana ((a ->5 b)' ^ a') = (a' ^ ((a v b) ^ (a v b')))
Proof of Theorem u5lemnana
StepHypRef Expression
1 u5lemoa 624 . . . 4 ((a ->5 b) v a) = (a v ((a' ^ b) v (a' ^ b')))
2 ax-a2 31 . . . . . 6 ((a' ^ b) v (a' ^ b')) = ((a' ^ b') v (a' ^ b))
3 anor3 90 . . . . . . . 8 (a' ^ b') = (a v b)'
4 anor2 89 . . . . . . . 8 (a' ^ b) = (a v b')'
53, 42or 72 . . . . . . 7 ((a' ^ b') v (a' ^ b)) = ((a v b)' v (a v b')')
6 oran3 93 . . . . . . 7 ((a v b)' v (a v b')') = ((a v b) ^ (a v b'))'
75, 6ax-r2 36 . . . . . 6 ((a' ^ b') v (a' ^ b)) = ((a v b) ^ (a v b'))'
82, 7ax-r2 36 . . . . 5 ((a' ^ b) v (a' ^ b')) = ((a v b) ^ (a v b'))'
98lor 70 . . . 4 (a v ((a' ^ b) v (a' ^ b'))) = (a v ((a v b) ^ (a v b'))')
101, 9ax-r2 36 . . 3 ((a ->5 b) v a) = (a v ((a v b) ^ (a v b'))')
11 oran 87 . . 3 ((a ->5 b) v a) = ((a ->5 b)' ^ a')'
12 oran1 91 . . 3 (a v ((a v b) ^ (a v b'))') = (a' ^ ((a v b) ^ (a v b')))'
1310, 11, 123tr2 64 . 2 ((a ->5 b)' ^ a')' = (a' ^ ((a v b) ^ (a v b')))'
1413con1 66 1 ((a ->5 b)' ^ a') = (a' ^ ((a v b) ^ (a v b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i5 48
This theorem is referenced by: (None)
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